For the second part, every element of HK can be uniquely written as hk as H ∩ K = {e}. Then the mapping HK → H × K given by hk → (h, k) is a homormorphism, as (h 1k 1)(h 2k 2) = h 1h 2k 1k 2→ (h 1h 2, k 1k 2) = (h 1, k 1)(h 2, k 2). This is one-to-one and onto, so it is an isomorphism.